Average Error: 0.0 → 0.0
Time: 2.0s
Precision: binary64
\[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)\]
↓
\[\log \left(\log 1 + \frac{1 + \sqrt{1 - x \cdot x}}{x}\right)\]
\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)
↓
\log \left(\log 1 + \frac{1 + \sqrt{1 - x \cdot x}}{x}\right)
(FPCore (x)
 :precision binary64
 (log (+ (/ 1.0 x) (/ (sqrt (- 1.0 (* x x))) x))))
↓
(FPCore (x)
 :precision binary64
 (log (+ (log 1.0) (/ (+ 1.0 (sqrt (- 1.0 (* x x)))) x))))
double code(double x) {
	return log((1.0 / x) + (sqrt(1.0 - (x * x)) / x));
}
↓
double code(double x) {
	return log(log(1.0) + ((1.0 + sqrt(1.0 - (x * x))) / x));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.0

    \[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)\]
  2. Using strategy rm

  3. Applied add-log-exp_binary64_148163.4

    \[\leadsto \log \left(\frac{1}{x} + \color{blue}{\log \left(e^{\frac{\sqrt{1 - x \cdot x}}{x}}\right)}\right)\]

  4. Applied add-log-exp_binary64_148163.4

    \[\leadsto \log \left(\color{blue}{\log \left(e^{\frac{1}{x}}\right)} + \log \left(e^{\frac{\sqrt{1 - x \cdot x}}{x}}\right)\right)\]

  5. Applied sum-log_binary64_153363.5

    \[\leadsto \log \color{blue}{\log \left(e^{\frac{1}{x}} \cdot e^{\frac{\sqrt{1 - x \cdot x}}{x}}\right)}\]

  6. Simplified63.5

    \[\leadsto \log \log \color{blue}{\left({\left(e^{\frac{1}{x}}\right)}^{\left(1 + \sqrt{1 - x \cdot x}\right)}\right)}\]
  7. Using strategy rm

  8. Applied *-un-lft-identity_binary64_144263.5

    \[\leadsto \log \log \left({\color{blue}{\left(1 \cdot e^{\frac{1}{x}}\right)}}^{\left(1 + \sqrt{1 - x \cdot x}\right)}\right)\]

  9. Applied unpow-prod-down_binary64_152163.5

    \[\leadsto \log \log \color{blue}{\left({1}^{\left(1 + \sqrt{1 - x \cdot x}\right)} \cdot {\left(e^{\frac{1}{x}}\right)}^{\left(1 + \sqrt{1 - x \cdot x}\right)}\right)}\]

  10. Applied log-prod_binary64_152863.5

    \[\leadsto \log \color{blue}{\left(\log \left({1}^{\left(1 + \sqrt{1 - x \cdot x}\right)}\right) + \log \left({\left(e^{\frac{1}{x}}\right)}^{\left(1 + \sqrt{1 - x \cdot x}\right)}\right)\right)}\]

  11. Simplified63.5

    \[\leadsto \log \left(\color{blue}{\log 1} + \log \left({\left(e^{\frac{1}{x}}\right)}^{\left(1 + \sqrt{1 - x \cdot x}\right)}\right)\right)\]

  12. Simplified0.0

    \[\leadsto \log \left(\log 1 + \color{blue}{\frac{1 + \sqrt{1 - x \cdot x}}{x}}\right)\]

  13. Final simplification0.0

    \[\leadsto \log \left(\log 1 + \frac{1 + \sqrt{1 - x \cdot x}}{x}\right)\]

Reproduce

herbie shell --seed 2020298 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1.0 x) (/ (sqrt (- 1.0 (* x x))) x))))